---
_id: '12683'
abstract:
- lang: eng
  text: We study the eigenvalue trajectories of a time dependent matrix Gt=H+itvv∗
    for t≥0, where H is an N×N Hermitian random matrix and v is a unit vector. In
    particular, we establish that with high probability, an outlier can be distinguished
    at all times t>1+N−1/3+ϵ, for any ϵ>0. The study of this natural process combines
    elements of Hermitian and non-Hermitian analysis, and illustrates some aspects
    of the intrinsic instability of (even weakly) non-Hermitian matrices.
acknowledgement: G. Dubach gratefully acknowledges funding from the European Union’s
  Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
  Grant Agreement No. 754411. L. Erdős is supported by ERC Advanced Grant “RMTBeyond”
  No. 101020331.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Guillaume
  full_name: Dubach, Guillaume
  id: D5C6A458-10C4-11EA-ABF4-A4B43DDC885E
  last_name: Dubach
  orcid: 0000-0001-6892-8137
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
citation:
  ama: Dubach G, Erdös L. Dynamics of a rank-one perturbation of a Hermitian matrix.
    <i>Electronic Communications in Probability</i>. 2023;28:1-13. doi:<a href="https://doi.org/10.1214/23-ECP516">10.1214/23-ECP516</a>
  apa: Dubach, G., &#38; Erdös, L. (2023). Dynamics of a rank-one perturbation of
    a Hermitian matrix. <i>Electronic Communications in Probability</i>. Institute
    of Mathematical Statistics. <a href="https://doi.org/10.1214/23-ECP516">https://doi.org/10.1214/23-ECP516</a>
  chicago: Dubach, Guillaume, and László Erdös. “Dynamics of a Rank-One Perturbation
    of a Hermitian Matrix.” <i>Electronic Communications in Probability</i>. Institute
    of Mathematical Statistics, 2023. <a href="https://doi.org/10.1214/23-ECP516">https://doi.org/10.1214/23-ECP516</a>.
  ieee: G. Dubach and L. Erdös, “Dynamics of a rank-one perturbation of a Hermitian
    matrix,” <i>Electronic Communications in Probability</i>, vol. 28. Institute of
    Mathematical Statistics, pp. 1–13, 2023.
  ista: Dubach G, Erdös L. 2023. Dynamics of a rank-one perturbation of a Hermitian
    matrix. Electronic Communications in Probability. 28, 1–13.
  mla: Dubach, Guillaume, and László Erdös. “Dynamics of a Rank-One Perturbation of
    a Hermitian Matrix.” <i>Electronic Communications in Probability</i>, vol. 28,
    Institute of Mathematical Statistics, 2023, pp. 1–13, doi:<a href="https://doi.org/10.1214/23-ECP516">10.1214/23-ECP516</a>.
  short: G. Dubach, L. Erdös, Electronic Communications in Probability 28 (2023) 1–13.
date_created: 2023-02-26T23:01:01Z
date_published: 2023-02-08T00:00:00Z
date_updated: 2023-10-17T12:48:10Z
day: '08'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1214/23-ECP516
ec_funded: 1
external_id:
  arxiv:
  - '2108.13694'
  isi:
  - '000950650200005'
file:
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  checksum: a1c6f0a3e33688fd71309c86a9aad86e
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  creator: dernst
  date_created: 2023-02-27T09:43:27Z
  date_updated: 2023-02-27T09:43:27Z
  file_id: '12692'
  file_name: 2023_ElectCommProbability_Dubach.pdf
  file_size: 479105
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file_date_updated: 2023-02-27T09:43:27Z
has_accepted_license: '1'
intvolume: '        28'
isi: 1
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 1-13
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
publication: Electronic Communications in Probability
publication_identifier:
  eissn:
  - 1083-589X
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Dynamics of a rank-one perturbation of a Hermitian matrix
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 28
year: '2023'
...
---
_id: '9230'
abstract:
- lang: eng
  text: "We consider a model of the Riemann zeta function on the critical axis and
    study its maximum over intervals of length (log T)θ, where θ is either fixed or
    tends to zero at a suitable rate.\r\nIt is shown that the deterministic level
    of the maximum interpolates smoothly between the ones\r\nof log-correlated variables
    and of i.i.d. random variables, exhibiting a smooth transition ‘from\r\n3/4 to
    1/4’ in the second order. This provides a natural context where extreme value
    statistics of\r\nlog-correlated variables with time-dependent variance and rate
    occur. A key ingredient of the\r\nproof is a precise upper tail tightness estimate
    for the maximum of the model on intervals of\r\nsize one, that includes a Gaussian
    correction. This correction is expected to be present for the\r\nRiemann zeta
    function and pertains to the question of the correct order of the maximum of\r\nthe
    zeta function in large intervals."
acknowledgement: The research of L.-P. A. is supported in part by the grant NSF CAREER
  DMS-1653602. G. D. gratefully acknowledges support from the European Union’s Horizon
  2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement
  No. 754411. The research of L. H. is supported in part by the Deutsche Forschungsgemeinschaft
  (DFG, German Research Foundation) through Project-ID 233630050 -TRR 146, Project-ID
  443891315 within SPP 2265 and Project-ID 446173099.
article_number: '2103.04817'
article_processing_charge: No
arxiv: 1
author:
- first_name: Louis-Pierre
  full_name: Arguin, Louis-Pierre
  last_name: Arguin
- first_name: Guillaume
  full_name: Dubach, Guillaume
  id: D5C6A458-10C4-11EA-ABF4-A4B43DDC885E
  last_name: Dubach
  orcid: 0000-0001-6892-8137
- first_name: Lisa
  full_name: Hartung, Lisa
  last_name: Hartung
citation:
  ama: Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta
    function over intervals of varying length. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2103.04817">10.48550/arXiv.2103.04817</a>
  apa: Arguin, L.-P., Dubach, G., &#38; Hartung, L. (n.d.). Maxima of a random model
    of the Riemann zeta function over intervals of varying length. <i>arXiv</i>. <a
    href="https://doi.org/10.48550/arXiv.2103.04817">https://doi.org/10.48550/arXiv.2103.04817</a>
  chicago: Arguin, Louis-Pierre, Guillaume Dubach, and Lisa Hartung. “Maxima of a
    Random Model of the Riemann Zeta Function over Intervals of Varying Length.” <i>ArXiv</i>,
    n.d. <a href="https://doi.org/10.48550/arXiv.2103.04817">https://doi.org/10.48550/arXiv.2103.04817</a>.
  ieee: L.-P. Arguin, G. Dubach, and L. Hartung, “Maxima of a random model of the
    Riemann zeta function over intervals of varying length,” <i>arXiv</i>. .
  ista: Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta
    function over intervals of varying length. arXiv, 2103.04817.
  mla: Arguin, Louis-Pierre, et al. “Maxima of a Random Model of the Riemann Zeta
    Function over Intervals of Varying Length.” <i>ArXiv</i>, 2103.04817, doi:<a href="https://doi.org/10.48550/arXiv.2103.04817">10.48550/arXiv.2103.04817</a>.
  short: L.-P. Arguin, G. Dubach, L. Hartung, ArXiv (n.d.).
date_created: 2021-03-09T11:08:15Z
date_published: 2021-03-08T00:00:00Z
date_updated: 2023-05-03T10:22:59Z
day: '08'
department:
- _id: LaEr
doi: 10.48550/arXiv.2103.04817
ec_funded: 1
external_id:
  arxiv:
  - '2103.04817'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2103.04817
month: '03'
oa: 1
oa_version: Preprint
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: arXiv
publication_status: submitted
status: public
title: Maxima of a random model of the Riemann zeta function over intervals of varying
  length
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '9281'
abstract:
- lang: eng
  text: We comment on two formal proofs of Fermat's sum of two squares theorem, written
    using the Mathematical Components libraries of the Coq proof assistant. The first
    one follows Zagier's celebrated one-sentence proof; the second follows David Christopher's
    recent new proof relying on partition-theoretic arguments. Both formal proofs
    rely on a general property of involutions of finite sets, of independent interest.
    The proof technique consists for the most part of automating recurrent tasks (such
    as case distinctions and computations on natural numbers) via ad hoc tactics.
article_number: '2103.11389'
article_processing_charge: No
arxiv: 1
author:
- first_name: Guillaume
  full_name: Dubach, Guillaume
  id: D5C6A458-10C4-11EA-ABF4-A4B43DDC885E
  last_name: Dubach
  orcid: 0000-0001-6892-8137
- first_name: Fabian
  full_name: Mühlböck, Fabian
  id: 6395C5F6-89DF-11E9-9C97-6BDFE5697425
  last_name: Mühlböck
  orcid: 0000-0003-1548-0177
citation:
  ama: Dubach G, Mühlböck F. Formal verification of Zagier’s one-sentence proof. <i>arXiv</i>.
    doi:<a href="https://doi.org/10.48550/arXiv.2103.11389">10.48550/arXiv.2103.11389</a>
  apa: Dubach, G., &#38; Mühlböck, F. (n.d.). Formal verification of Zagier’s one-sentence
    proof. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2103.11389">https://doi.org/10.48550/arXiv.2103.11389</a>
  chicago: Dubach, Guillaume, and Fabian Mühlböck. “Formal Verification of Zagier’s
    One-Sentence Proof.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2103.11389">https://doi.org/10.48550/arXiv.2103.11389</a>.
  ieee: G. Dubach and F. Mühlböck, “Formal verification of Zagier’s one-sentence proof,”
    <i>arXiv</i>. .
  ista: Dubach G, Mühlböck F. Formal verification of Zagier’s one-sentence proof.
    arXiv, 2103.11389.
  mla: Dubach, Guillaume, and Fabian Mühlböck. “Formal Verification of Zagier’s One-Sentence
    Proof.” <i>ArXiv</i>, 2103.11389, doi:<a href="https://doi.org/10.48550/arXiv.2103.11389">10.48550/arXiv.2103.11389</a>.
  short: G. Dubach, F. Mühlböck, ArXiv (n.d.).
date_created: 2021-03-23T05:38:48Z
date_published: 2021-03-21T00:00:00Z
date_updated: 2023-05-03T10:26:45Z
day: '21'
department:
- _id: LaEr
- _id: ToHe
doi: 10.48550/arXiv.2103.11389
ec_funded: 1
external_id:
  arxiv:
  - '2103.11389'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2103.11389
month: '03'
oa: 1
oa_version: Preprint
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: arXiv
publication_status: submitted
related_material:
  record:
  - id: '9946'
    relation: other
    status: public
status: public
title: Formal verification of Zagier's one-sentence proof
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '10285'
abstract:
- lang: eng
  text: We study the overlaps between right and left eigenvectors for random matrices
    of the spherical ensemble, as well as truncated unitary ensembles in the regime
    where half of the matrix at least is truncated. These two integrable models exhibit
    a form of duality, and the essential steps of our investigation can therefore
    be performed in parallel. In every case, conditionally on all eigenvalues, diagonal
    overlaps are shown to be distributed as a product of independent random variables
    with explicit distributions. This enables us to prove that the scaled diagonal
    overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail
    limit, namely, the inverse of a γ2 distribution. We also provide formulae for
    the conditional expectation of diagonal and off-diagonal overlaps, either with
    respect to one eigenvalue, or with respect to the whole spectrum. These results,
    analogous to what is known for the complex Ginibre ensemble, can be obtained in
    these cases thanks to integration techniques inspired from a previous work by
    Forrester & Krishnapur.
acknowledgement: We acknowledge partial support from the grants NSF DMS-1812114 of
  P. Bourgade (PI) and NSF CAREER DMS-1653602 of L.-P. Arguin (PI). This project has
  also received funding from the European Union’s Horizon 2020 research and innovation
  programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. We would
  like to thank Paul Bourgade and László Erdős for many helpful comments.
article_number: '124'
article_processing_charge: No
article_type: original
author:
- first_name: Guillaume
  full_name: Dubach, Guillaume
  id: D5C6A458-10C4-11EA-ABF4-A4B43DDC885E
  last_name: Dubach
  orcid: 0000-0001-6892-8137
citation:
  ama: Dubach G. On eigenvector statistics in the spherical and truncated unitary
    ensembles. <i>Electronic Journal of Probability</i>. 2021;26. doi:<a href="https://doi.org/10.1214/21-EJP686">10.1214/21-EJP686</a>
  apa: Dubach, G. (2021). On eigenvector statistics in the spherical and truncated
    unitary ensembles. <i>Electronic Journal of Probability</i>. Institute of Mathematical
    Statistics. <a href="https://doi.org/10.1214/21-EJP686">https://doi.org/10.1214/21-EJP686</a>
  chicago: Dubach, Guillaume. “On Eigenvector Statistics in the Spherical and Truncated
    Unitary Ensembles.” <i>Electronic Journal of Probability</i>. Institute of Mathematical
    Statistics, 2021. <a href="https://doi.org/10.1214/21-EJP686">https://doi.org/10.1214/21-EJP686</a>.
  ieee: G. Dubach, “On eigenvector statistics in the spherical and truncated unitary
    ensembles,” <i>Electronic Journal of Probability</i>, vol. 26. Institute of Mathematical
    Statistics, 2021.
  ista: Dubach G. 2021. On eigenvector statistics in the spherical and truncated unitary
    ensembles. Electronic Journal of Probability. 26, 124.
  mla: Dubach, Guillaume. “On Eigenvector Statistics in the Spherical and Truncated
    Unitary Ensembles.” <i>Electronic Journal of Probability</i>, vol. 26, 124, Institute
    of Mathematical Statistics, 2021, doi:<a href="https://doi.org/10.1214/21-EJP686">10.1214/21-EJP686</a>.
  short: G. Dubach, Electronic Journal of Probability 26 (2021).
date_created: 2021-11-14T23:01:25Z
date_published: 2021-09-28T00:00:00Z
date_updated: 2021-11-15T10:48:46Z
day: '28'
ddc:
- '519'
department:
- _id: LaEr
doi: 10.1214/21-EJP686
ec_funded: 1
file:
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  checksum: 1c975afb31460277ce4d22b93538e5f9
  content_type: application/pdf
  creator: cchlebak
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  date_updated: 2021-11-15T10:10:17Z
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  file_size: 735940
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file_date_updated: 2021-11-15T10:10:17Z
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intvolume: '        26'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: Electronic Journal of Probability
publication_identifier:
  eissn:
  - 1083-6489
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: On eigenvector statistics in the spherical and truncated unitary ensembles
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
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  short: CC BY (4.0)
type: journal_article
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
volume: 26
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...